An eigenstate formulation of the magnetotelluric impedance tensor

Abstract

An important step in the interpretation of magnetotelluric (MT) data is the extraction of scalar parameters from the impedance tensor Z, the transfer function which relates the observed horizontal magnetic and electric fields. The conventional approach defines parameters in terms of elements of a coordinate‐rotated tensor. The rotation angle is chosen such that Z′(θ) approximates in some sense the form for a two‐dimensional (2-D) subsurface conductivity distribution, with zero elements on the diagonal. There are two major problems with this approach. (1) Apparent resistivities, defined from the off‐diagonal elements of the rotated tensor, are independent of the trace of Z. It is problematic that apparent resistivities, the parameters for which we have physical analogs and which are most heavily used in interpretation, are insensitive to the addition of an arbitrary constant on the diagonal of Z. (2) The conventional parameter set is incomplete; there are two degrees of freedom in Z which are transparent to all parameters. Through a variation of the classical eigenstate formulation of a matrix, it is shown that in general there exist two, and only two, polarization states for which the electric and magnetic fields have the same polarization at perpendicular orientations. For each eigenstate the magnetic and electric fields are related by a scalar, the eigenvalue for that state. This scalar relationship between fields is of identical form to the solution for transverse electromagnetic (TEM) waves in a homogeneous medium and thus provides a physically more satisfactory basis for defining apparent resistivity than the conventional approach using the off‐diagonal elements of the coordinate‐rotated impedance tensor. The eigenstate and coordinate‐rotation methods yield identical results in the limited cases of 1-D and 2-D subsurface conductivity distributions. The eigenstates provide the basis for new definitions of parameters as concise, closed expressions which are complete and more amenable to interpretational insight. The polarization ellipses defined by the eigenstates provide a concise display in real space of all the information contained in the impedance tensor.